What is the integral of 1/(sqrt(-x^2+14x+50)) ?

The integral of 1/(sqrt(-x^2+14x+50)) is arcsin(1/(3sqrt(11))(x-7))+C

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integral of 1/(sqrt(-x^2+14x+50))

\int \frac{1}{- x ^{2} + 14 x + 50} d x

arcsin (\frac{1}{3 11} (x - 7)) + C

Solution steps

\int \frac{1}{- x ^{2} + 14 x + 50} d x

Complete the square

- x^{2} + 14 x + 50 : - (x - 7)^{2} + 99

= \int \frac{1}{- ( x - 7 ) ^{2} + 99} d x

Apply u-substitution

: \int \frac{1}{- u ^{2} + 99} d u

= \int \frac{1}{- u ^{2} + 99} d u

Apply Trigonometric Substitution

: \int 1 d v

= \int 1 d v

Integral of a constant:

\int a d x = a x

= 1 \cdot v

Substitute back

= 1 \cdot arcsin (\frac{1}{3 11} (x - 7))

Simplify

= arcsin (\frac{1}{3 11} (x - 7))

Add a constant to the solution

= arcsin (\frac{1}{3 11} (x - 7)) + C

1 + sin^{2} (e^{5 x})

\frac{d}{d x} (ln (x + x^{2} + 3))

2 x (1 + y) d x - y d y = 0

x^{8} ln (x)

f (x) = e^{3 x}

What is the integral of 1/(sqrt(-x^2+14x+50)) ?
The integral of 1/(sqrt(-x^2+14x+50)) is arcsin(1/(3sqrt(11))(x-7))+C